Poisson Process Partition Calculus with applications to Exchangeable models and Bayesian Nonparametrics

TL;DR

This paper develops a partition-based Fubini calculus for Poisson processes, extending Bayesian nonparametric models and deriving new formulas for Levy-Cox, Poisson-Dirichlet, and Neutral to the Right processes.

Contribution

It introduces an explicit partition calculus for inhomogeneous models, including exponential change of measure formulas, advancing Bayesian nonparametric theory.

Findings

Derived explicit partition calculus for complex models

Established new identities for Poisson-Dirichlet and Levy-Cox processes

Extended Markov-Krein correspondence and Neutral to the Right calculus

Abstract

This article discusses the usage of a partiton based Fubini calculus for Poisson processes. The approach is an amplification of Bayesian techniques developed in Lo and Weng for gamma/Dirichlet processes. Applications to models are considered which all fall within an inhomogeneous spatial extension of the size biased framework used in Perman, Pitman and Yor. Among some of the results; an explicit partition based calculus is then developed for such models, which also includes a series of important exponential change of measure formula. These results are applied to obtain results for Levy-Cox models, identities related to the two-parameter Poisson-Dirichlet process and other processes, generalisations of the Markov-Krein correspondence, calculus for extended Neutral to the Right processes, among other things.